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On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

  • * Corresponding author: Barbara Kaltenbacher

    * Corresponding author: Barbara Kaltenbacher 

Supported by the Austrian Science Fund fwf under grant P30054 and the National Science Foundation through award dms-1620138

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  • We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as $ B/A $ in the literature which is part of a space dependent coefficient $ \kappa $ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set $ \Sigma $ representing the receiving transducer array. After an analysis of the map from $ \kappa $ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $ \kappa $. Numerical simulations will also be shown to illustrate the efficiency of the methods.

    Mathematics Subject Classification: Primary: 35R30, 35K58, 35L72; Secondary: 78A46.


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  • Figure 1.  The surface Σ

    Figure 2.  Reconstructions of a smooth $ \kappa(x) $ from time trace data at $ \,x = 1\, $ under 0.1% (left) and 1% (right) noise using Newton's method

    Figure 3.  Reconstructions of piecewise linear $ \kappa(x) $ from time trace data at $ \,x = 1 $ under 0.1% (left) and 1% (right) noise using Newton's method

    Figure 4.  Reconstructions of a piecewise constant $ \kappa(x) $ from time trace data at $ x = 1 $ under $ 0.1\% $ noise using Newton iteration

    Figure 5.  Comparison of Newton (in red) and Halley (in blue) final reconstructions under $ 0.1\% $ noise

    Figure 6.  Comparison of Newton (in red) and Halley (in blue) final reconstructions and norm differences of the $ n^{\rm th} $ iterate $ \kappa_n $ and the actual $ \kappa $. Noise level was $ 1\% $

    Figure 7.  Reconstructions of a piecewise linear $ \kappa(x) $ from time trace data at $ x = 1 $ under $ 1\% $ noise using Landweber iteration

    Figure 8.  The leftmost figure shows reconstructions of $ \kappa(x) $ under $ 0.1\% $ noise using Landweber iteration. The rightmost figure shows the decay of the norm $ \kappa_n(x)-\kappa_{\rm act}(x) $

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